The title of this book might well suggest to the average reader a class on philosophical ethics, but in fact it is hard-core algebraic geometry. I sometimes wonder whether Grothendieck, when he came up with the word “motive,” already had in mind the many puns that it allows. It is clear that he originally meant a “motive” in the musical sense, i.e., a musical phrase or theme that can be heard repeatedly and recognized. But, at least in English, the psychological meaning of the word is much more prevalent, and when an adjective like “pure” or “mixed” is attached to the word it is hard to resist that interpretation. (I have heard that a mathematician — who shall remain unnamed — once wrote a paper with the sole purpose of defining an “ulterior motive.”) Since the term “pure motive” goes back to Grothendieck himself, I guess he is the one to blame.
So what are these motives? The idea goes back to the 1960s. If one wants to study an algebraic variety defined, say, by a set of polynomial equations with rational coefficients, one of the best tools is cohomology. But which cohomology? By the 1960s, one had several at hand: the classical cohomology theory of complex manifolds, DeRham cohomology, and étale cohomology. The étale theory could be considered over any p-adic field, yielding an infinite number of different cohomologies.
One of the most important results of the period was a comparison theorem that showed that in fact these theories were very closely related. While some of these cohomologies carried various kinds of “extra structure,” at base they were all quite similar. This led Grothendieck to dream of some underlying structure, a common motive underlying all of these musical/cohomological expressions.
Motives turned out to be a very fruitful and inspirational idea. Mathematicians working in arithmetic algebraic geometry talk about them all the time. But pinning them down precisely and proving theorems about them has turned out to be quite hard.
Grothendieck formulated quite precisely what these motives should look like. They should carry the basic information that would then be expressed in various “realizations” of the motive, which would be the various cohomology theories. And he described how to create such a theory in the simplest case, the “pure motives” described in this book. There still no completely satisfactory theory of the more general “mixed motives” Even for pure motives the theory is not completely in place: we have a good construction, but in order to prove that they do what they are supposed to do one would need to have a proof of the “standard conjectures” about algebraic cycles. Those are far from settled.
The lectures in this book were given by Murre, originally in 1988 and then many other times in various settings. With the help of Nagel and Peters, they have been written up completely and can now be made available to all who want to know more. They cover the construction and theory of pure motives, but also offer some hints about how to go beyond them. They are, of course, quite technical, but those of us who want to add some more detail to the vague “philosophy of motives” will welcome them gladly.
Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.